Lemma 4.8 Suppose that \(\beta ^{A}\)  is shear-free. Then the following sequence
is a differential complex if and only if \(\beta ^{A}\)  is also a repeated principal spinor of the ASD Weyl
curvatureA similar complex was obtained in [1]}, but not in the context of Lie algebroids.:
\(0 \rightarrow \Gamma (\Lambda ^{0}_{(r,r^{\prime };w)}) \xrightarrow{} \Gamma (\Lambda ^{1}_{(r,r^{\prime };w)})\xrightarrow{} \Gamma (\Lambda ^{2}_{(r,r^{\prime };w)}) \rightarrow 0.\) 

